the other, like as we also reckon the vibrations of a pendulum." Holden adds that Dr. Smith, in his *Harmonics*, reckons the complete vibration to be double of this. Lees, in his *Acoustics*, says- "The travel of a vibrating elastic body from one extreme to the opposite and back again is called a vibration. Continental writers define a vibration to be the travel of a vibrating body from one extreme position to the opposite. This corresponds to our definition of the oscillation of a pendulum."

These definitions have been given for strings and pendulums have been wrong for both. Indeed, the vibration of a string is not even once from extreme to extreme; for while the string itself goes from one extreme to the other, it moves in half the time of one vibration and half the time of another. In the first half of its course, that is from the extreme to the right line, or line of its rest, it leaves the air on that side to expand itself; and in the second half of its course, that is from the right line to the other extreme, it compresses the air on the other side. Now, a vibration of a string does not consist in the expanding of a body of air on the one side of a string and the compression of a different body of air on the other side, but in the compression and expansion of the same body of air on the one side of the string; so the vibration of the string are on either side of the line of its rest. The vibration is the movement from the right line, or center of action, to the extreme and back again to the right line, and so on the other side.

This definition of a vibration answers to all the requirements of the case; and in exhibiting the ratio 1:2 with two strings, it brings the two strings into the same position at every second vibration of the higher one. And so with every ratio of the musical system.

As vibrating motions of whatever kind, and however rapid, are made in time, so they all partake of its continuity. Where the one vibration ends the other begins, at an indivisible instant when the string is crossing the right line. An indivisible point is the common limit of the two vibrations; thus they are under the law of continuity.

The contraction and expansion of the air in one vibration is effected in successive times; but the compression of the air of one vibration and the expansion of the air of the preceding one are simultaneous, so they